Principia mathematica, by Alfred North Whitehead ... and Bertrand Russell.

*212. THE SERIES OF SEGMENTS.

Summary of *212.
The series of segments or sections of a series may be ordered by the
relation of inclusion, after the manner considered in *210. Since, as was
shown in *211, sections and segments have the properties assigned to Ic in
the hypothesis of *210, the resulting series are such that every class has
either a maximum or a sequent, and either a minimum or a precedent;
i.e. the series of segments or sections are Dedekindian. Most of the
properties of the series of sections and of the series of segments which have
no maximum, only require that the original relation should be connected.
The properties of the series of segments in general (D'PE) require also that
the original relation should be transitive.
We denote the series of segments by s'P, putting
S'P=Pc D'Pe Df.
We then have, in virtue of *210'13 and *211-61,
*212'23. F: P e trans n connex. D. 'P = a/, {a, / e D'P. a C. a 4+ /3
In like manner, for the series of segments which have no maximum,
we put
sgm'P = Pe D'(Pe n I) Df,
and we have
*212-22.: P e connex.. sgm'P = a,8 {a, e D'(Pe A I). a C /3. a + }
We do not need a special notation for the series of sections, since, in virtue
of *211-13, it is s'P* or sgm'P*. Thus, by *212-23,
*212-24. F: Pe connex. D.;'P* = a/3 a, 3 e sect'P. a C 3. a +,}
We begin the number with various propositions on the fields, etc. of these
relations, and on the conditions for their existence. We have
*212-132. F. D' 'P = D'P, - 'D'P. C's'P = D'Pe - t'A
*212'133. F: g! P. ). C's'P = D'Pe. B's'P = A. B'Cnv''P = D'P
*212-14. F: P.. g! ' P
*212-152. F. (I'sgm'P = D'(Pe A I)- t'A